Pendulum

[Ussuri]

I have a 7000kg pendulum on the end of a 5m long wire. The pendulum will be subjected to a lateral load of 62kN. Is there a quick method of detemining maximum amplitude or do I need to go back to harmonic motion and determine the equations from scratch?



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[zekeman]

Ussuri,
I found this for site for the harmonic analysis:

http://math.ut.ee/~toomas_l/harmonic_analysis/

You need the Fourier series harmonics, which will yield
Ak, Bk components where your roll function is represented as
A0+sum AkSin(2pikt/T) +BkCos(2pikt/T).

As far as the solution to the diff equation, any second year book on differential equations has it, or go online to get it. For your equation, there are two solutions, one is the homogeneous where the diff equation has no forcing function and the other called the particular solution. For your problem you need only the latter whose solution is (I previously gave it in another form)
Wi^2/(W^2-Wi^2){(Ak)sin(kWit)+ (Bk)cos(kWit)}
where
Wi is the fundamental frquency of the roll and k is the kth harmonic
W is the natural frquency of the pendulum=Sqrt(g/l)

 

[GregLocock]

Well, worst case, seems to be that the hook remains at the same height, so Pythagoras gives you the answer.

In other words, there is not much energy coupling between a forcing function with a fundamental at 1/13 Hz, and a pendulum at say 2 Hz. That does not seem unreasonable.

If you come up with a better description of the geometry, I can refine the estimate. I just guessed to meet what you had stated. The most useful missing numbers would be the peak to peak lateral amplitude of the motion of the suspension point on the crane, and its height above the centre of rotation.





Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[swearingen]

I know smaller marine cranes don't normally use them, but the larger ones I worked with (500 to 5000 tons) used air tuggers for load control. Why not just tie the ball/load down and be done with it...

 

[Cockroach]

This problem is one of Elliptic Integrals (First Kind), the 62 kN side load gives an initial deflection of the 7000 kg bob greater than 10 degrees. Thus the linear extrapolation of the sine function in the differential equation of motion is greater than a single term.

You may wish to consult the classical textbook, Elliptic Integrals, Hancock 1917, pg 90, example 7. I obtained a copy of the original textbook off eBay several years ago.

Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada